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PreprintarXiv licencearXiv

Limits of Hyperkähler Manifolds and Moduli Spaces of Sheaves on K3 Surfaces

  • Kefeng Liu
  • Yang Shen
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Abstract

We prove that a compact strongly Gauduchon manifold which is a limit of compact hyperkähler manifolds is bimeromorphic to a compact hyperkähler manifold. This settles a case of a conjecture of Soldatenkov and Verbitsky. A closed completion of a Gauduchon metric bounds the Dirichlet energies of nearby biholomorphisms. This bound yields a meromorphic limit map, which has degree one and is therefore a bimeromorphism onto a compact hyperkähler manifold. We apply this result to moduli spaces of stable torsion-free coherent sheaves on K3 surfaces. A Bott--Chern diagonal formula gives the ∂∂̅-lemma, and positivity of determinant classes gives Kählerness, proving Perego's conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026limits,
  title = {{Limits of Hyperkähler Manifolds and Moduli Spaces of Sheaves on K3 Surfaces}},
  author = {Kefeng Liu and Yang Shen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.18065},
  eprint = {2605.18065},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.18065},
}
RIS
TY  - UNPB
TI  - Limits of Hyperkähler Manifolds and Moduli Spaces of Sheaves on K3 Surfaces
AU  - Kefeng Liu
AU  - Yang Shen
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2605.18065
UR  - https://arxiv.org/abs/2605.18065
AB  - We prove that a compact strongly Gauduchon manifold which is a limit of compact hyperkähler manifolds is bimeromorphic to a compact hyperkähler manifold. This settles a case of a conjecture of Soldatenkov and Verbitsky. A closed completion of a Gauduchon metric bounds the Dirichlet energies of nearby biholomorphisms. This bound yields a meromorphic limit map, which has degree one and is therefore a bimeromorphism onto a compact hyperkähler manifold. We apply this result to moduli spaces of stable torsion-free coherent sheaves on K3 surfaces. A Bott--Chern diagonal formula gives the ∂∂̅-lemma, and positivity of determinant classes gives Kählerness, proving Perego's conjecture.
ER  -